{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:W5XCDYKH4ERMIG4MVL65EIHHMN","short_pith_number":"pith:W5XCDYKH","canonical_record":{"source":{"id":"2501.12055","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-01-21T11:26:38Z","cross_cats_sorted":[],"title_canon_sha256":"2bfce1a7dcb3a0579354645e3b0428ef8e82788f6b5cf3443a68105d0b2ed6ed","abstract_canon_sha256":"85bf81f76ee7d42cd6d5c8cae58eb54551da7048a883e7069e021d8904510ca1"},"schema_version":"1.0"},"canonical_sha256":"b76e21e147e122c41b8caafdd220e763731d2bb15f0eaca4ec4ade0f48e93bb1","source":{"kind":"arxiv","id":"2501.12055","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.12055","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"2501.12055v1","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.12055","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"W5XCDYKH4ERM","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_16","alias_value":"W5XCDYKH4ERMIG4M","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_8","alias_value":"W5XCDYKH","created_at":"2026-07-05T10:03:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:W5XCDYKH4ERMIG4MVL65EIHHMN","target":"record","payload":{"canonical_record":{"source":{"id":"2501.12055","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-01-21T11:26:38Z","cross_cats_sorted":[],"title_canon_sha256":"2bfce1a7dcb3a0579354645e3b0428ef8e82788f6b5cf3443a68105d0b2ed6ed","abstract_canon_sha256":"85bf81f76ee7d42cd6d5c8cae58eb54551da7048a883e7069e021d8904510ca1"},"schema_version":"1.0"},"canonical_sha256":"b76e21e147e122c41b8caafdd220e763731d2bb15f0eaca4ec4ade0f48e93bb1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:03:31.224848Z","signature_b64":"OocjPgSeMP/ajJVz4tzSHSSlkfPF2LZ9lMWNthcPmapxjdeV21zU+IUflAMX8vLrF3z+jiB4kVby2P9SxMRIBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b76e21e147e122c41b8caafdd220e763731d2bb15f0eaca4ec4ade0f48e93bb1","last_reissued_at":"2026-07-05T10:03:31.224455Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:03:31.224455Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2501.12055","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8nUFdi1qI5OAwpnsaeygtfWIfswrzEsHc2bPCud5EyTDxLRlGjxXhzN5B+Bw8LP+tyLC3zhHrmxxWHLYku5NCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:47:57.460174Z"},"content_sha256":"5832d508c0ca113a7e8a13eb62e7b563d6c9afe386fbfa9846ff3922b8f97202","schema_version":"1.0","event_id":"sha256:5832d508c0ca113a7e8a13eb62e7b563d6c9afe386fbfa9846ff3922b8f97202"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:W5XCDYKH4ERMIG4MVL65EIHHMN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorics on bi-$\\gamma$-positivity of $1/k$-Eulerian polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sherry H.F. Yan, Xubo Yang, Zhicong Lin","submitted_at":"2025-01-21T11:26:38Z","abstract_excerpt":"The $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-$\\gamma$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$\\gamma$-coefficients still remains open. The study of the theme of bi-$\\gamma$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$\\gamma$-coefficients of $A^{(k)}_{n}(x)$ by using the model of certain ordered"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.12055","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.12055/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WDgJH1We4hZLi+fXlBNdgjZwCH4eiV8/cDuLPyAvg1X3PFVYWD98LXrMa5X+0TMxj/a4fcjyyoWDOwOn140hCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:47:57.460723Z"},"content_sha256":"cabaca415b5ae1b7cb1cf7f2752ea172d9923130af0094bc7e3874e914f1dde1","schema_version":"1.0","event_id":"sha256:cabaca415b5ae1b7cb1cf7f2752ea172d9923130af0094bc7e3874e914f1dde1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/bundle.json","state_url":"https://pith.science/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T18:47:57Z","links":{"resolver":"https://pith.science/pith/W5XCDYKH4ERMIG4MVL65EIHHMN","bundle":"https://pith.science/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/bundle.json","state":"https://pith.science/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W5XCDYKH4ERMIG4MVL65EIHHMN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:W5XCDYKH4ERMIG4MVL65EIHHMN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"85bf81f76ee7d42cd6d5c8cae58eb54551da7048a883e7069e021d8904510ca1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-01-21T11:26:38Z","title_canon_sha256":"2bfce1a7dcb3a0579354645e3b0428ef8e82788f6b5cf3443a68105d0b2ed6ed"},"schema_version":"1.0","source":{"id":"2501.12055","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.12055","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"2501.12055v1","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.12055","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"W5XCDYKH4ERM","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_16","alias_value":"W5XCDYKH4ERMIG4M","created_at":"2026-07-05T10:03:31Z"},{"alias_kind":"pith_short_8","alias_value":"W5XCDYKH","created_at":"2026-07-05T10:03:31Z"}],"graph_snapshots":[{"event_id":"sha256:cabaca415b5ae1b7cb1cf7f2752ea172d9923130af0094bc7e3874e914f1dde1","target":"graph","created_at":"2026-07-05T10:03:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2501.12055/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-$\\gamma$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$\\gamma$-coefficients still remains open. The study of the theme of bi-$\\gamma$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$\\gamma$-coefficients of $A^{(k)}_{n}(x)$ by using the model of certain ordered","authors_text":"Sherry H.F. Yan, Xubo Yang, Zhicong Lin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-01-21T11:26:38Z","title":"Combinatorics on bi-$\\gamma$-positivity of $1/k$-Eulerian polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.12055","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5832d508c0ca113a7e8a13eb62e7b563d6c9afe386fbfa9846ff3922b8f97202","target":"record","created_at":"2026-07-05T10:03:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"85bf81f76ee7d42cd6d5c8cae58eb54551da7048a883e7069e021d8904510ca1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-01-21T11:26:38Z","title_canon_sha256":"2bfce1a7dcb3a0579354645e3b0428ef8e82788f6b5cf3443a68105d0b2ed6ed"},"schema_version":"1.0","source":{"id":"2501.12055","kind":"arxiv","version":1}},"canonical_sha256":"b76e21e147e122c41b8caafdd220e763731d2bb15f0eaca4ec4ade0f48e93bb1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b76e21e147e122c41b8caafdd220e763731d2bb15f0eaca4ec4ade0f48e93bb1","first_computed_at":"2026-07-05T10:03:31.224455Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:03:31.224455Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OocjPgSeMP/ajJVz4tzSHSSlkfPF2LZ9lMWNthcPmapxjdeV21zU+IUflAMX8vLrF3z+jiB4kVby2P9SxMRIBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:03:31.224848Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.12055","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5832d508c0ca113a7e8a13eb62e7b563d6c9afe386fbfa9846ff3922b8f97202","sha256:cabaca415b5ae1b7cb1cf7f2752ea172d9923130af0094bc7e3874e914f1dde1"],"state_sha256":"b8614bb3cc63de945440736e50906e10805429b2b808c970382398c4200df868"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2YExi6Br7hr75zTAfN5xZBeUlyElh+LbT5v6U29jf+Cir8coMnRAbrNbQRFCKsSJpIIXQzob+NNPqrSGRT/8Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T18:47:57.465472Z","bundle_sha256":"b772be702b04e4a528c7e48bfb3ec1dba125f59b712b28a56e4839d0dad5d868"}}